Math 111 Section 4.6 #29 and #31

S. Silver

#29) "Your friend is using Descartes's Rule of Signs to find the number of negative real roots of x^3+x^2+x+1=0. Describe and correct the error."

My Friend's Work and Answer:

"P(-x) = (-x)^3 +(-x)^2 + (-x) +1

= -x^3 - x^2 - x +1"

"Because there is only on sign change in P(-x), there must be one negative real root."

Here is my work:

The error that my friend made was that they kept a negative x^2 instead of changing it to a positive x^2. My friend also made an error with how many sign changes there are, they said that it was only one sign change when there are actually three sign changes. There is only one negative real root, which my friend got right.

Then this is the graphs of x^3+x^2+x+1=0 and its one negative real root, which was -1.

#31) A gardener is designing a new garden in the shape of a trapezoid. She wants the shorter base to be twice the height and the longer base to be 4 feet longer than the shorter base. If she has enough topsoil to create a 60ft^2 garden, what dimensions should she use for her garden?

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